The addition formula for Jacobi polynomials and spherical harmonics : prepublication

Recently Gegenbauer’s addition formula was generalized for Jacobi polynomials $P_n^{( {\alpha ,\beta } )} ( x )$ by an algebraic approach (cf. the author’s paper in [10]). Here a new algebraic proof using spherical harmonics will be presented. Let $q\leqq p$ and let $e_1 ,e_2 , \cdots ,e_{q + p} $ be an orthonormal basis of $R^{q + p} $ with unit sphere $\Omega $. The Jacobi polynomials $P_n^{( p/2- 1,q/2 - 1)} ( x )$ can be characterized as spherical harmonics of degree $2n$ on $\Omega $ which are invariant under the subgroup $SO( q ) \times SO( p )$ of the rotation group $SO( {q + p} )$. Let the rotations $A_\tau $ be defined by $A_\tau e_k = \cos \tau e_k - \sin \tau e_{q + k} ,k = 1, \cdots ,q,A_\tau e_{q + k} = \sin \tau e_k + \cos \tau e_{q + k} ,k = 1, \cdots ,q,A_\tau e_k = e_k ,k = 2q + 1, \cdots ,q + p$. An explicit orthonormal basis will be constructed for the set of those spherical harmonics of degree $2n$ which are invariant under all $T \in SO( q ) \times SO( p )$ which commute with the rota...